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The quadratic function and quadratic equations
1985The function f(x), where f(x) = ax2 + bx + c, and a, b, c are constants, a ≠ 0, is called a quadratic function, or sometimes a quadratic polynomial. From elementary algebra $${(x + d)^2} \equiv {x^2} + 2dx + {d^2}.$$ Using this, we write $$a{x^2} + bx + c \equiv a\left( {{x^2} + \frac{b}{a}x + \frac{c}{a}} \right) \equiv a\left[ {{{\left( {x
C. Plumpton, J. E. Hebborn
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Quadratic Operators and Quadratic Functional Equation
2012In the first part of this paper, we consider some quadratic difference operators (e.g., Lobaczewski difference operators) and quadratic-linear difference operators (d’Alembert difference operators and quadratic difference operators) in some special function spaces X λ . We present results about boundedness and find the norms of such operators.
S. Czerwik, M. Adam
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Fuzzy stability of the cubic and quadratic functional equation
, 2016In this paper, we investigate a fuzzy version of stability for the functional equation f(x+ 2y)− 2f(x+ y) + 2f(x− y)− f(x− 2y)− f(2y) + 4f(−y) = 0 in the sense of Mirmostafaee and Moslehian.
Yang-Hi Lee, Soon-Mo Jung
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On stability of a functional equation of quadratic type
Acta Mathematica Hungarica, 2016We prove some stability results for the equation $$Af(px * ry) + Bf(qx * sy) = Cf(x) + Df(y),$$ in the class of functions mapping a groupoid (X, ∗) into a Banach space Y , where \({p, q, r, s: X \rightarrow X}\) are endomorphisms of the groupoid, and A, B, C, D are fixed scalars.
Mohammad Sal Moslehian +3 more
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Quadratic Functional Equations
2009Quadratic functional equations, bilinear forms equivalent to the quadratic equation, and some generalizations are treated in this chapter. Among the normed linear spaces (n.l.s.), inner product spaces (i.p.s.) play an important role. The interesting question when an n.l.s. is an i.p.s. led to several characterizations of i.p.s.
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Set-Valued Quadratic Functional Equations
Results in Mathematics, 2017In this paper, we introduce set-valued quadratic functional equations and prove the Hyers–Ulam stability of the set-valued quadratic functional equations by using the fixed point method.
Choonkil Park +3 more
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Quadratic Functional Equations
2011So far, we have discussed the stability problems of functional equations in connection with additive or linear functions. In this chapter, the Hyers–Ulam–Rassias stability of quadratic functional equations will be proved. Most mathematicians may be interested in the study of the quadratic functional equation since the quadratic functions are applied to
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Quadratic functions satisfying an additional equation
Acta Mathematica Hungarica, 2020There is a result, due independently to Kurepa [14] and to Jurkat [12], which distinguishes linear functions or derivations from other additive functions as solutions to certain functional equations. The purpose of this paper is to prove an analogue of a part of this result, corresponding to derivations, for quadratic functions.
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Quadratic variation functionals and dilation equations
Potential Analysis, 1995Here we present some distribution function inequalities between certain functionals defined relative to a convolution approximation procedure. Such inequalities are best known when the approximation is made using dilations of the Gaussian or Cauchy kernels.
Ileana Iribarren, R. F. Gundy
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On the quadratic functional equation on groups
2004We study the solutions f: G→H of the quadratic functional equation on G, where G and H are groups, H abelian. We show that any solution f is a function on the quotient group [G,[G,G]]. By help of this we find sufficient conditions on G for all solutions to satisfy Kannappan's condition.
Friis, P.d.P., Stetkær, H.
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